96 XI! TOPOLOGY AND TOPOLOGICAL ALGEBRA 25. (a) Give an example of a sequence (/) of continuous real-valued functions on an interval I c R which converges simply at each point of I but is such that there exists no nonempty open interval contained in I on which the sequence converges uniformly (cf. (12.7.1)). (b) Let (/) be a sequence of complex analytic functions on an open set D c C which converges simply at each point of D. Show that there exists a dense open set U in D such that, for each z0 e U, there exists a neighborhood of z0 on which the sequence (/) is uniformly convergent. (Consider the sequence of functions zh-» \fn(z)\ and use (9.13.2).) 26. Let t\-^A(t) be a continuous mapping of the interval R+ = [0, + oo[ of R into the space of real n x n matrices, and let t*-+C(t) be the solution of the homogeneous linear differential equation U' = A(t)U which is equal to the unit matrix when t = 0 (10.8.4). Let EI denote the vector subspace of Rn consisting of vectors y such that the function t\-~+C(t) - y is bounded on [0, + oo[, and let E2 be a supplement of EX in R". Let PI : R" -* Ex and P2 : Rn -> E2 be the projections associated with the direct- sum decomposition R" = Ex © E2 (5.4). (a) Suppose that, for each function fe <tf£n(R+), the differential equation Kr A(t) - x +/(0 has at least one solution which is bounded on R+ . Show that, for each function / e ^°n (R+), there exists a unique vector y(f) e E2 such that the unique solution uf of the equation x' = A(t) - x + f(t) which takes the value y(f) at the point 0 is bounded in R+ , and that there exists a constant c > 0 such that (Consider the space F of continuously differentiate functions x on R+ with values in R" such that x and x' A x are bounded on R+ and #(0) e E2. Show that the function is a norm on F, with respect to which F is a Banach space. Then apply Banach's theorem to the linear mapping jti>x' A -xofF into #£, (R+), bearing in mind the definition of E2.) (b) Deduce from (a) that a necessary and sufficient condition for the equation x' = A(t) - x+f(t) to have a solution which is bounded on R+, for all functions sup ( f \\C(t)P± C"1(s)\\ ds + f ||C(t)P2C'l(s) || ds] < + oo t>o \Jo J r / (cf. (10.8.6) and (13.14.4)). 27. Let/: [a,£>[~»R be an increasing function, continuous on the right. Suppose that the countable set of points of discontinuity of / is dense in [a, b[ (Section 3.15, Problem). Show that the set of points x e [a, b[ at which /has a finite derivative on the right is meager. (For each integer n > 0, let An be the set of points x e [a, b[ such that there exist points y, z e [a, b[ satisfying x < y < z < x + l//zand /Cy)-/(*) /(*)-/(*) y x z x Show that AM is a dense open subset of [a, b[.)t, for each sequence (£) of real